Financial markets can support the transition to a low-carbon economy by redirecting funds from highly emissive to clean investments. We study whether European stock markets incorporate carbon prices in company valuations and to what degree they discriminate between firms with different carbon intensities. Using a novel dataset of stock prices and carbon intensities of 338 European publicly traded companies between 2013 and 2021, we find a strongly statistically significant relationship between weekly carbon price changes and stock returns. Crucially, this relationship depends on firms’ carbon intensity: the higher the carbon costs a firm faces, the poorer its stock performance during the periods of carbon price increases. Emissions covered with free allowances however do not affect this relationship, illustrating how both carbon pricing and disclosures are needed for financial markets to foster climate change mitigation. The relationship we identify can provide an incentive for firms to decarbonize. We argue in favor of more ambitious carbon pricing policies, as this would strengthen the stock-market incentive channel while causing only limited financial stability risk for stocks.